If there are just a few burns, can you steal a small section from a closet area or perhaps under the fridge to replace the damaged section.
While paint may work, You would need to do the entire floor to hide the damage. And paint normally does not adhere all that well to a vinyl surface.
Depending on the size of the damage, get a short piece of copper tubing, file the outer edge carefully to create a sharp edge. Set it over the damage and whack it with a hammer. Then the same with the area that is unoticed as explained above. Glue the good piece in the old hole. There will always be a ring where dirt will collect where circle was cut though. It's unavoidable as it is a patch.
They make special glues for patches that melts the vinyl so there is not gaps between the joints. Ideally you need to remove a small section if you have it hidden under a fridge or closet then place it over the offending spot and using a sharp utility knife cut out both sections at the same time so they are exactly the same size. If done properly combined with the pattern being the same and properly glued, you should never see it.